Convert to an equality.
syms x_1 x_2 x_3 x_4 x_max real
syms delta
assume(delta>0);
eqns= [3* x_1 * (x_max)^2 + 2 * x_2 * x_max + x_3 == 0, ...
1/4 * 100^4 * x_1 + 1/3 * 100^3 * x_2 + 1/2 * 100^2 * x_3 + 100 * x_4 == 1, ...
5^3 * x_1 + 5^2 * x_2 + 5 *x_3 + x_4 == 0, ...
100^3 * x_1 + 100^2 * x_2 + 100 * x_3 + x_4 == 2.3 + delta ];
S = solve(eqns, [x_1 x_2 x_3 x_4 x_max]);
x_1, x_2, and x_4 will come out in terms of delta, with x_3 and x_max coming out 0.
You can then make delta positive and arbitrarily close to 0 or as large and positive as you want
>> subs([S.x_1 S.x_2 S.x_3 S.x_4 S.x_max], delta, 10)
ans =
[ 697378134818815959/14008751663786663333750, -42103925179940864/11207001331029330667, 0, 9828603160166400041/112070013310293306670, 0]
>> subs(eqns, [x_1 x_2 x_3 x_4 x_max], ans)
ans =
[ 0 == 0, 1 == 1, 0 == 0, 123/10 == delta + 23/10]
The last of those expressions shows you that the value that would be calculated by the left side of the inequality would be 123/10, which is greater than 23/10 on the right hand side of the inequality, with the difference being the 10 that was substituted for delta